A099040 → x ↦ x−15

Proved for all n: all roots real, in (13, 15] · alternating coefficient signs · interlacing (weak) what this means

Recurrence

\(P_{n}(x) = \left(-30 + 2\,x\right)\,P_{n-1}(x) + \left(-30 + 2\,x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = -30 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	-30	2														-28
2	870	-118	4													756
3	-25200	5160	-352	8												-20384
4	729900	-199920	20524	-936	16											549584
5	-21141000	7252200	-994680	68184	-2336	32										-14817600
6	612333000	-252390600	43329240	-3965752	204096	-5600	64									399504448
7	-17735760000	8536536000	-1760313600	201596160	-13847936	570560	-13056	128								-10771231744
8	513702810000	-282771216000	68077821600	-9362880960	804576016	-44236480	1519680	-29824	256							290408364288
9	-14879011500000	9218974500000	-2537994600000	407473560000	-42044412000	2891433760	-132530560	3904128	-67072	512						-7829839711232
10	430959260700000	-296816715900000	91969909920000	-16883153928000	2033416437600	-167895590368	9624720960	-378250880	9753088	-148992	1024					211104077714432
11	-12482407476000000	9460092740400000	-3258152942400000	673134241680000	-92692521504000	8932868749440	-614774025216	30214783360	-1039273984	23826432	-327680	2048				-5691678664089600
12	361543446459000000	-299001177165600000	113312043023400000	-26019898697520000	4032275327496000	-444267404904960	35684425445824	-2105394582912	90558691840	-2769364992	57154560	-714752	4096			153456088418504704
13	-10471831169490000000	9384354610722000000	-3880698871280400000	980510713837200000	-168881013091440000	20939201696649600	-1922758614929280	132394696827776	-6835942134784	261404992512	-7195883520	135034880	-1548288	8192		-4137403473123622912
14	303308631690930000000	-292781178452754000000	131192311714822800000	-36169498110704400000	6854443763197680000	-944545504280227200	97602094267992960	-7682827446312832	462940107778048	-21249835711488	731433123840	-18307061760	314966016	-3334144	16384	111550526771743309824

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Technical details

Node id: n-22b2bd96c59e3f63 · revision 1 · JSON
Definition source: derived:basis_map_coefficients_of:oeis_catalog:A099040
Fingerprint: f6-3dd98d96997242cb

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): no
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

The atlas's own rows: n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:               1;
  1:             -30,             2;
  2:             870,          -118,              4;
  3:          -25200,          5160,           -352,            8;
  4:          729900,       -199920,          20524,         -936,           16;
  5:       -21141000,       7252200,        -994680,        68184,        -2336,         32;
  6:       612333000,    -252390600,       43329240,     -3965752,       204096,      -5600,         64;
  7:    -17735760000,    8536536000,    -1760313600,    201596160,    -13847936,     570560,     -13056,     128;
  8:    513702810000, -282771216000,    68077821600,  -9362880960,    804576016,  -44236480,    1519680,  -29824,    256;
  9: -14879011500000, 9218974500000, -2537994600000, 407473560000, -42044412000, 2891433760, -132530560, 3904128, -67072, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(30-2*x)*P(n-1,x) - (30-2*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = -30 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := -30 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(30 - 2*x)*P[n - 1, x] - (30 - 2*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = -30 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(30 - 2*x)*v[i - 1] - (30 - 2*x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Generated with polytool c48629374597.

Proved properties

Real-rooted. Every row P_n, n >= 0, is zero or real-rooted with all roots in (13, 15]
Every polynomial of the sequence has only real roots, inside the stated interval.

by the cited steps: e-48e4dfa7d694f880
Coefficients of one sign. Every nonzero row, n >= 0: the coefficients of P_n(-x) have one sign
All coefficients of each polynomial have the same sign (after x -> -x when alternating).

derived from the cited steps by a stated theorem: e-48e4dfa7d694f880
Interlacing of consecutive rows. Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
The roots of consecutive polynomials alternate on the real line.

by the cited steps: e-48e4dfa7d694f880

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

preserver_image affine, shift 0 e-48e4dfa7d694f880

Affine, reversal and Möbius substitutions (elementary) polytool ?
- premise (sequence): Riordan array (1, 2+2x).
  uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-2, 0]
  - theorem_check ma_wang_interval e-17786677d2ed021b root: theorem check
    
    Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ?

Used in 1 proof step

e-0a457ba83443136e basis_map (A027467) → Riordan array (1, 2+2x).

Related sequences ?

No candidate equalities or related sequences.

References

[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741